Optimal. Leaf size=82 \[ \frac {\sqrt {a^2 x^2+1} x^{m+1} F_1\left (m+1;\frac {1}{2} (5-i n),\frac {1}{2} (i n+5);m+2;i a x,-i a x\right )}{c^2 (m+1) \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.21, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5085, 5082, 133} \[ \frac {\sqrt {a^2 x^2+1} x^{m+1} F_1\left (m+1;\frac {1}{2} (5-i n),\frac {1}{2} (i n+5);m+2;i a x,-i a x\right )}{c^2 (m+1) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 5082
Rule 5085
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)} x^m}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{n \tan ^{-1}(a x)} x^m}{\left (1+a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int x^m (1-i a x)^{-\frac {5}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {5}{2}-\frac {i n}{2}} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^{1+m} \sqrt {1+a^2 x^2} F_1\left (1+m;\frac {1}{2} (5-i n),\frac {1}{2} (5+i n);2+m;i a x,-i a x\right )}{c^2 (1+m) \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {e^{n \tan ^{-1}(a x)} x^m}{\left (c+a^2 c x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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